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4 SPECTRUM OF A COMPACT OPERATOR 323

PROBLEMS

1. Let E, F be two Banach spaces, /a continuous linear mapping of E into F such that
/(E) = F; then, there exists a number m > 0 such that for any y e F, there is an x e E
for which/(*) = y and ||*|| < m\\y\\ (12,16.12).

(a) If (y_n) is a sequence of points of F which converges to a point b_y show that there
exists a subsequence (y_nk), and a sequence (x_k) of points of E, which converges to a
point a and is such that f(x_k) = y_nk for every k. (Take (y_nk) such that the series of
general term \\y_nk+i - y_nk\\ is convergent.)

(b) Let u be a compact mapping of E into F, and let v = /— u. Show that v(E) is
closed in F and has finite codimension in F. (Follow the same pattern as in the proof
of (11.3.2), using (a).)

(c) Define inductively F_A = t?(E), F_k+1 = v(f'¹(F_k)) for k ^ I; show that there is an
integer n such that F_fc+i = F_fc for k ^ n (same method).

(d) Take E=Fto be a separable Hilbert space, and let (e_n)_n&i be a Hilbert
basis of E. Define/and u such that f(e_n) = e_n-s for n ^ 4,f(e_n) = 0 for n ^ 3, nfe) =
£_n_₂/fl for n ^ 6, «(<?i) = w(<?₃) = 0, «(e₂) = — e₂, «(£*) — <?i, w(e_s) = <?₂ -f G?s/5). Define
inductively N! = iT^O), N^ «= t^H/CN*)) for k ^ 1; show that the N_fc are all distinct
and finite dimensional.

2. Let E, F be two normed spaces, /a linear homeomorphism of E onto a closed subspace
/(E) of F, u a compact mapping of E into F, and let v = /— u.

(a) Show that v~^l(Q) is finite dimensional and v(E) is closed in F; furthermore, if
y-i(O) = {0}, t;is a linear homeomorphism of E onto v(E). (Follow the same method
as in (11.3.2).)

(b) Define inductively NI = t?"^a(0), N*+i = ^(/(N*)) for k ^ 1; show that there is
an integer n such that N/t₊₁ = N_fc for k > n.

(c) Give an example in which, when FI = v(E), and F_fc + i = v(f~*(F_k)) for k ^ 1,
the F_fc are all distinct (take for E = F a separable Hilbert space, and for / and u the
adjoints (Section 11.5) of the mappings noted/and u in Problem l(d)).

3. Let E be a Banach space, g a continuous linear mapping of E onto itself such that
||#||<J; then f=^\E — g is a linear homeomorphism of E onto itself (8.3.2.1).
Let u be a compact operator in E, and let v=f— «; then the statements in (11.3.2)
and (11.3.3) are all valid. (First prove the following result, corresponding to (11.3.1):
if M <= L, M =£ L and v(L) c M, there is an a E L n C M such that \\a\\ ^ 1 and for any
x e M such that \\x\\ ^ 1, \\u(a) - u(x)\\ ^ (1 - 2||#||)/2.)

4. In the space E = I¹ (Section 5.7, Problem 1; we keep the notations of that problem),
let/ be the automorphism of E such that/(<?₂k) = <?2*+2 (k ^ 0),/(^) = <?₀, /(<?2k+i) =
e2_fc_i for k^\₉ and let u be the compact mapping such that u(e_n) = 0 for n ^ 1, and
u(_ei) = e₀. If v —f— u, and the F_fc and N_fc are defined as in (11.3.3), show that N*₊i ^
N_k and F_fc+l ^ F_fc for every k.

4. SPECTRUM OF A COMPACT OPERATOR

(11.4.1) Let u be a compact operator in a complex normed space E. Then:
(a) The spectrum Sofu is an at most denumerable compact subset of C,
each point of which, with the possible exception o/O, is isolated; 0 belongs to S
ifE is infinite dimensional.



	4 SPECTRUM OF A COMPACT OPERATOR 323

	PROBLEMS 1. Let E, F be two Banach spaces, /a continuous linear mapping of E into F such that /(E) = F; then, there exists a number m > 0 such that for any y e F, there is an x e E for which/() = y* and \|\|\|\| < m\\y\\* (12,16.12). (a) If (y_n) is a sequence of points of F which converges to a point b_y show that there exists a subsequence (y_nk), and a sequence (x_k) of points of E, which converges to a point a and is such that f(x_k) = y_nk for every k. (Take (y_nk) such that the series of general term \\y_nk+i - y_nk\\ is convergent.) (b) Let u be a compact mapping of E into F, and let v = /— u. Show that v(E) is closed in F and has finite codimension in F. (Follow the same pattern as in the proof of (11.3.2), using (a).) (c) Define inductively F_A = t?(E), F_k+1 = v(f'¹(F_k)) for k ^ I; show that there is an integer n such that F_fc+i = F_fc for k ^ n (same method). (d) Take E=Fto be a separable Hilbert space, and let (e_n)_n&i be a Hilbert basis of E. Define/and u such that f(e_n) = e_n-s for n ^ 4,f(e_n) = 0 for n ^ 3, nfe) = £_n_₂/fl for n ^ 6, «(<?i) = w(<?₃) = 0, «(e₂) = — e₂, «(£) — <?i, w(e_s) = <?₂ -f G?s/5). Define inductively N! = iT^O), N^ «= t^H/CN)) for k ^ 1; show that the N_fc are all distinct and finite dimensional. 2. Let E, F be two normed spaces, /a linear homeomorphism of E onto a closed subspace /(E) of F, u a compact mapping of E into F, and let v = /— u. (a) Show that v~^l(Q) is finite dimensional and v(E) is closed in F; furthermore, if y-i(O) = {0}, t;is a linear homeomorphism of E onto v(E). (Follow the same method as in (11.3.2).) (b) Define inductively NI = t?"^a(0), N+i = ^(/(N)) for k ^ 1; show that there is an integer n such that N/t₊₁ = N_fc for k > n. (c) Give an example in which, when FI = v(E), and F_fc + i = v(f~(F_k))* for k ^ 1, the F_fc are all distinct (take for E = F a separable Hilbert space, and for / and u the adjoints (Section 11.5) of the mappings noted/and u in Problem l(d)). 3. Let E be a Banach space, g a continuous linear mapping of E onto itself such that \|\|#\|\|<J; then f=^\E — g is a linear homeomorphism of E onto itself (8.3.2.1). Let u be a compact operator in E, and let v=f— «; then the statements in (11.3.2) and (11.3.3) are all valid. (First prove the following result, corresponding to (11.3.1): if M <= L, M =£ L and v(L) c M, there is an a E L n C M such that \\a\\ ^ 1 and for any x e M such that \\x\\ ^ 1, \\u(a) - u(x)\\ ^ (1 - 2\|\|#\|\|)/2.) 4. In the space E = I¹ (Section 5.7, Problem 1; we keep the notations of that problem), let/ be the automorphism of E such that/(<?₂k) = <?2+2 (k ^ 0),/(^) = <?₀, /(<?2k+i) = e2_fc_i for k^\₉* and let u be the compact mapping such that u(e_n) = 0 for n ^ 1, and u(_ei) = e₀. If v —f— u, and the F_fc and N_fc are defined as in (11.3.3), show that N*₊i ^ N_k and F_fc+l ^ F_fc for every k.

	4. SPECTRUM OF A COMPACT OPERATOR (11.4.1) Let u be a compact operator in a complex normed space E. Then: (a) The spectrum Sofu is an at most denumerable compact subset of C, each point of which, with the possible exception o/O, is isolated; 0 belongs to S ifE is infinite dimensional.